# Intuitive idea behind orbit stabilizer theorem.

Recently I studied the orbit stabilizer theorem which is as follows:

Suppose $$G$$ is a group acting on $$X$$ (i.e.$$X$$ is a $$G$$-set). Let $$x\in X$$,then define, $$\operatorname {orb}(x):=\{g.x:g\in G\}$$ and $$\operatorname{stab}(x)=\{g\in G:g.x=x\}$$, then we have $$|\operatorname{orb}(x)|=[G:\operatorname{stab}(x)]$$, provided $$G$$ and $$X$$ are finite.

Now I have tried to intuitively understand this theorem like this, orbit of $$x$$ is roughly speaking, all the possible points in $$X$$ where $$x$$ can go under the given group action. And stabilizer means all the group elements(we can think of them as permutations also) that fix $$x$$.

Now the index on the right hand side of orbit stabilizer theorem is the number of cosets of $$G$$ induced by the subgroup viz stabilizer of $$x$$. Whenever a subgroup induces a coset or partition within a group, it means that we are classifying the group element by discriminating the group elements based on the propertly of that subgroup, now the property of stabilizer of $$x$$ is that it fixes $$x$$, so it contains all elements of $$G$$ that fix $$x$$, so the other coset must be according to the property,where the group elements take $$x$$, all the members of the group taking $$x$$ to a specific point will be a coset and for every point in orbit, there is a coset of $$G$$ whose members take $$x$$ to that point.

So there is a one-one correspondence between the cosets of $$G$$ induced by stabilizer subgroup and the members of the orbit of $$x$$,notice that $$x$$ itself is a member of the orbit and for it,the corresponding coset is stabilizer of $$x$$ itself. So,naturally the number of orbit elements is the same as the number of cosets of stabilizer of $$x$$ in $$G$$.

I want to know if my intuition or understanding correct?

• @Berci can you give me suitable diagram. Feb 2, 2020 at 13:42
• Not a diagram, but you can draw one based on my answer. Feb 2, 2020 at 13:48
• I've always been a big fan of this blog post by Timothy Gowers. It starts off with a slightly more concrete intuitive example of orbit-stabiliser, and goes on to formalise this intuition. Perhaps you will find it helpful. Feb 2, 2020 at 13:59
• @IzaakvanDongen I wanted to know one more thing $Ker\phi=\cap_{x\in X} stab(x)$ right?where $ker\phi$ is the kernel of $\phi : G \to Sym(X)$ defined by $\phi(g)=\sigma_g$ where $\sigma_g(x)=(g,x)$ where $(.,.)$ is a group action of $G$ on $X$. Feb 2, 2020 at 14:02
• Yes, in order to be in $\operatorname{Ker} \phi$ you should fix everything, and hence be in every stabiliser, and conversely if you are in every stabiliser then you fix everything so your permutation representation is the identity permutation. Feb 2, 2020 at 14:04

The (left) cosets of $$S:=\mathrm{stab}(x)$$ correspond bijectively to the elements in the orbit of $$x$$:
Specifically, for any elements $$g,h\in G$$, we have $$gx=hx\ \iff\ h^{-1}gx=x\ \iff\ h^{-1}g\in S\ \iff\ gS=hS\,.$$